Abstract
Sum rules for structure functions and their twist-2 relations have important roles in constraining their magnitudes and x dependencies and in studying higher-twist effects. The Wandzura-Wilczek (WW) relation and the Burkhardt-Cottingham (BC) sum rule are such examples for the polarized structure functions g1 and g2. Recently, new twist-3 and twist-4 parton distribution functions were proposed for spin-1 hadrons, so that it became possible to investigate spin-1 structure functions including higher-twist ones. We show in this work that an analogous twist-2 relation and a sum rule exist for the tensor-polarized parton distribution functions f1LL and fLT, where f1LL is a twist-2 function and fLT is a twist-3 one. Namely, the twist-2 part of fLT is expressed by an integral of f1LL (or b1) and the integral of the function f2LT = (2/3)fLT− f1LL over x vanishes. If the parton-model sum rule for f1LL (b1) is applied by assuming vanishing tensor-polarized antiquark distributions, another sum rule also exists for fLT itself. These relations should be valuable for studying tensor-polarized distribution functions of spin-1 hadrons and for separating twist-2 components from higher-twist terms, as the WW relation and BC sum rule have been used for investigating x dependence and higher-twist effects in g2. In deriving these relations, we indicate that four twist-3 multiparton distribution functions FLT, GLT, {H}_{LL}^{perp } , and HTT exist for tensor-polarized spin-1 hadrons. These multiparton distribution functions are also interesting to probe multiparton correlations in spin-1 hadrons. In the near future, we expect that physics of spin-1 hadrons will become a popular topic, since there are experimental projects to investigate spin structure of the spin-1 deuteron at the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility, the electron-ion colliders in US and China in 2020’s and 2030’s.
Highlights
Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility, the electronion colliders in US and China in 2020’s and 2030’s
We show in this work that an analogous twist-2 relation and a sum rule exist for the tensorpolarized parton distribution functions f1LL and fLT, where f1LL is a twist-2 function and fLT is a twist-3 one
We indicate that four twist-3 multiparton distribution functions FLT, GLT, HL⊥L, and HT T exist for tensor-polarized spin-1 hadrons
Summary
There are useful sum rules and relations among them for finding their functional behavior. There is a useful relation for the polarized structure function g2, which exists in the spin-1/2 nucleons. The polarized distribution functions g1L, gT , and g3L are defined by the matrix element of a nonlocal operator as d(P +ξ−) eixP +ξ− 2π. The structure functions are classified by the twist, which is defined by the mass dimension minus spin, in the operator product expansion [32,33,34,35,36,37,38]. It is written by the following operators with the gluon field tensor Gμν, the quark mass (mq), and equation of motion as [32]. If tensor-polarized antiquark distributions vanish, there is a sum rule for the twist-2 collinear structure function as dxb1(x) = 0. We show the existence of a new sum rule and a twist-2 relation, which are analogous to the BC sum rule and the WW relation, respectively
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